# A Neural Network Approach for High-Dimensional Optimal Control Applied to Multi-Agent Path Finding

arXiv: https://arxiv.org/abs/2104.03270 · HTML: https://mlsanity.com/paper/2104.03270/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2021.

## Built on by (most cited first, 18 of 18)

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- [Systematic construction of continuous-time neural networks for linear dynamical systems](https://mlsanity.com/paper/2403.16215/) (2024)
- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)
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- [Zero-Shot Function Encoder-Based Differentiable Predictive Control](https://mlsanity.com/paper/2511.05757/) (2025)
- [End-to-End Learning of Safe Optimal Feedback Control in High Dimensions with Control Barrier Function Layers](https://mlsanity.com/paper/2607.20674/) (2026)

## Abstract (arXiv)

We propose a neural network approach that yields approximate solutions for high-dimensional optimal control problems and demonstrate its effectiveness using examples from multi-agent path finding. Our approach yields controls in a feedback form, where the policy function is given by a neural network (NN). Specifically, we fuse the Hamilton-Jacobi-Bellman (HJB) and Pontryagin Maximum Principle (PMP) approaches by parameterizing the value function with an NN. Our approach enables us to obtain approximately optimal controls in real-time without having to solve an optimization problem. Once the policy function is trained, generating a control at a given space-time location takes milliseconds; in contrast, efficient nonlinear programming methods typically perform the same task in seconds. We train the NN offline using the objective function of the control problem and penalty terms that enforce the HJB equations. Therefore, our training algorithm does not involve data generated by another algorithm. By training on a distribution of initial states, we ensure the controls' optimality on a large portion of the state-space. Our grid-free approach scales efficiently to dimensions where grids become impractical or infeasible. We apply our approach to several multi-agent collision-avoidance problems in up to 150 dimensions. Furthermore, we empirically observe that the number of parameters in our approach scales linearly with the dimension of the control problem, thereby mitigating the curse of dimensionality.